$K_{\ell 4}$ Form-Factors and $π$-$π$ Scattering

TL;DR

This work advances CHPT to ${ m O}(p^6)$ for $K_{\ell 4}$ decays in the three-flavour sector, delivering a full two-loop calculation of the $F$ and $G$ form-factors and updating the low-energy constants through a global fit. It systematically separates contributions into ${ m O}(p^6)$ pieces from loops, counterterms, and resonance/saturation inputs, and it analyzes isospin-related data and partial-wave content to connect $K_{\ell 4}$ with ${\pi}$-${\pi}$ scattering. The study also quantifies vacuum condensates and pseudoscalar masses/decay constants at this order, highlighting sizeable ${ m O}(p^6)$ effects and the sensitivity to resonance estimates and the large-$N_c$ assumptions. The results provide a consistent framework for interpreting current and upcoming $K_{\ell 4}$ data and for cross-checking CHPT predictions against low-energy hadron phenomenology. Overall, the paper strengthens the link between precise kaon decays, meson scattering, and the CHPT parameterization of QCD's chiral dynamics at low energies.

Abstract

The $F$ and $G$ form-factors of $K_{\ell4}$ and the quark condensates are calculated to ${\cal O}(p^6)$ in Chiral Perturbation Theory (CHPT). Full formulas are presented as much as possible. A full refit of most of the ${\cal O}(p^4)$ CHPT parameters is done with a discussion of all inputs and underlying assumptions. We discuss the consequences for the vacuum expectation values, decay constants, pseudoscalar masses and $π$-$π$ scattering.

Figures (16)

• Figure 1: Kinematical variables in $K_{\ell 4}$ decays.
• Figure 2: (a) One-particle irreducible tree level diagram. (b) One-particle irreducible one-loop diagrams. Dots refer to strong vertices or current insertions from ${\cal L}_2$, ${\cal L}_4$ or ${\cal L}_6$. External legs stand for pseudoscalar or weak current. Internal lines are pseudoscalars only.
• Figure 5: Fit to the $F_V$ and $G_V$ contribution. Plotted are the fit with lines and the full calculation with +. In all cases we plotted $\cos\theta_\pi=0,\pm1$ and $s_\ell=0,0.01,0.0225$ GeV$^2$.
• Figure 6: The sets of $L_i^r$ within a 68% confidence level range of $\chi^2$. In the first plot we show + the values of the first three $L_i^r$ and $\times$ the projections on the coordinate planes. The second plot shows the $L_7^r$--$L_8^r$ correlations.
• Figure 7: Comparison of the Omnès improved estimate with the full ${\cal O}(p^6)$ calculation using the same old set of values of $L_i^r$ as input together with $F_\pi=93.2$ MeV. The shaded band is the experimental result.